The Annals of Statistics

Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence

Iain M. Johnstone

Full-text: Open access

Abstract

Let A and B be independent, central Wishart matrices in p variables with common covariance and having m and n degrees of freedom, respectively. The distribution of the largest eigenvalue of (A+B)−1B has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that m and n grow in proportion to p. We show that after centering and scaling, the distribution is approximated to second-order, O(p−2/3), by the Tracy–Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role.

Article information

Source
Ann. Statist. Volume 36, Number 6 (2008), 2638-2716.

Dates
First available: 5 January 2009

Permanent link to this document
http://projecteuclid.org/euclid.aos/1231165182

Digital Object Identifier
doi:10.1214/08-AOS605

Mathematical Reviews number (MathSciNet)
MR2485010

Zentralblatt MATH identifier
05503373

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 62E20: Asymptotic distribution theory 15A52

Keywords
Canonical correlation analysis characteristic roots Fredholm determinant Jacobi polynomials largest root Liouville–Green multivariate analysis of variance random matrix theory Roy’s test soft edge Tracy–Widom distribution

Citation

Johnstone, Iain M. Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence. The Annals of Statistics 36 (2008), no. 6, 2638--2716. doi:10.1214/08-AOS605. http://projecteuclid.org/euclid.aos/1231165182.


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